Optimal. Leaf size=86 \[ -\frac {c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{b x \left (a+b \text {sech}^{-1}(c x)\right )} \]
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Rubi [A] time = 0.14, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6285, 3297, 3303, 3298, 3301} \[ -\frac {c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{b x \left (a+b \text {sech}^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6285
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b}\\ &=\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {\left (c \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b}+\frac {\left (c \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{b}\\ &=\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}+\frac {c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 82, normalized size = 0.95 \[ \frac {-c \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+c \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x \left (a+b \text {sech}^{-1}(c x)\right )}}{b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arsech}\left (c x\right ) + a^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 164, normalized size = 1.91 \[ c \left (\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{2 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{2 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{2 b^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c^{2} x^{3} + {\left (c^{2} x^{3} - x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - x}{{\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{2} \log \relax (x) + {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) + a b\right )} x^{2} - {\left (b^{2} x^{2} \log \relax (x) + {\left (b^{2} \log \relax (c) - a b\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + {\left (\sqrt {c x + 1} \sqrt {-c x + 1} b^{2} x^{2} - {\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{2}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )} + \int -\frac {c^{4} x^{4} - 2 \, c^{2} x^{2} - {\left (c^{2} x^{2} + 1\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - {\left (c^{2} x^{2} - 2\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + 1}{{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{2} \log \relax (x) - {\left (b^{2} x^{2} \log \relax (x) + {\left (b^{2} \log \relax (c) - a b\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left ({\left (b^{2} c^{4} \log \relax (c) - a b c^{4}\right )} x^{4} - 2 \, {\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} + b^{2} \log \relax (c) - a b\right )} x^{2} - 2 \, {\left ({\left (b^{2} c^{2} x^{2} - b^{2}\right )} x^{2} \log \relax (x) + {\left ({\left (b^{2} c^{2} \log \relax (c) - a b c^{2}\right )} x^{2} - b^{2} \log \relax (c) + a b\right )} x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} + {\left ({\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} x^{2} + 2 \, {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{2} - {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} x^{2}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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